Hopf quasicomodules and Yetter-Drinfel'd quasicomodules

Let H be a Hopf coquasigroup over a field k possessing an adjoint quasicoaction. We first show that if M is any right H-module and N is any right H-quasicomodule such that , where is a favorable map, then we have H = k. As an application of this result, we get that symmetric category of Yetter-Drinfeld quasicomodules over H is trivial, as a generalization of Pareigis' Theorem. Furthermore, let (H, R) be a quasitriangular Hopf coquasigroup and coquasitriangular Hopf coquasigroup. Then, we show that the category of generalized Long quasicomodules is a braided monoidal subcategory of Yetter-Drinfeld category . Finally, we give a new approach to a braided monoidal category by generalizing one of Schauenburg's main results in the setting of Hopf coquasigroups introduced by Klim and Majid. This yields new sources of braidings that provide solutions to the Yang-Baxter equation playing an important role in various areas of mathematics.